Electronic Absorptions of C6o: A Quantum-Mechanical Model David W. Ball Cleveland State University, Cleveland, OH 44115 The recent synthesis (1)of Cso,buckminsterfullerene, has provided scientists with a totally new molecular shape. Although in reality the structure is icosahedral (21, its beautiful three-dimensional symmetry suggests that this molecule might be approximated as a sphere. If the rr: electrons are assumed to be distributed over the entire molecule, the changes in the energy levels of the electrons should be predicted by the expression for the energies of a particle confined to rotation in three dimensions: a 3-D rigid rotor. To date, this comparison has not been pointed out. The W-vis (i.e., electronic) spectrum of Cm has been reand comparison of theory and ported several times (1,3,4), experiment shows that this analogy is indeed accurate. The quantum-mechanical treatment of the three-dimensional rigid rotor, given in many elementary and graduatelevel textbooks (51, indicates that the energy of a particle confined to a spherical surface having radius r and moment of inertia I = m? is
The energy is dependent on the angular momentum quantum number I. The selection rules for the allowed changes in energy levels of a 3-D rigid rotor are Thus, it is possible to determine AE for an allowed transition. Using the mass of the electron and the known (6) diameter of the hllerene sphere, 7.0 A, the change in energy is
Comparison of Measured Electronic Absorptions of CEO with Predictions based on Rigid-Rotor Approximation Measured Energy, nm ( I , 3, 4)
Calculated Energy, nm
Transition, Rigid Rotor Approx.
est transition, based on this model, would be expected for 1 = 4 to I = 5. The calculated wavelength for this transition, using eq 3 above, is 398 nm;an experimentally measured transition occurs at 404 nm (1,3,4). The table shows that this approximation is not coincidence; a 3-D rigid rotor model also predicts accurately the 1 = 5 + I = 6 transition as well as a 1 = 7 +1 = 8 transition. This last fitting is peculiar because it is expected that no level above 2 = 5 would be populated thermally at room temperature. A Deslandres-type table of AE's for all levels 1 = 0 through 2 = 8 shows some close fits for Al being different than 1,but none fit as well as the three listed in the table. It remains to be seen if selection rules are so lax in this system that formally forbidden transitions have a transition moment large enough to justify the assignment of transitions to Al = 2, 3,4 . . . The assignments given here are notable for their simplicity of explanation using the 3-D rigid rotor model. They are certainly worth consideration as both a guide to further theoretical description as well as a potentially valuable pedagogical tool (7). Literature Cited
where 1 is the quantum number of the lower angular momentum state. (AE is expressed in terms of wavelength, nm,because this unit is used in the reported UV-vis spectra (1,3,4).) Given 60 rr: electrons and an overall degeneracy of 2(21+ 11,all angular momentum levels are filled up to the 1 = 5 level, which is only partially filled. Therefore, the low-
1. Kratachmer, W; Lamb, L.: Foatiropoulos. K,Huffmsn, DNoflrn 1890,347,354 2. Johnsm,R.D.:Meijer, G.:Bethune,D. S.J. Am. Cham. Soc Ism,112,8933. 3. Ajie.H.;Alvsrez.M. M.;Ane, S. J.;Bptk, S. D.;Diedel?ch, F.:Foetiropmioa, K.:Huffman, D . R.: Kratsehmer, W:Rubin,Y:Sehriver K,SenBarma, D.; Whetten,R. L. J Phys. Chem. 1990.94.8610. 4. Hare, J. P,Kmto,H.W;Taylor, R.Chom.Phyn. Lett. 1631,177,394. 5. Lwine. 1.N. Quantum Chomlstry,4thed.:PrenticrHall: Englmmd CWa, NJ, 1991: Levhe. I. N.Physico1 Chemistry; McGraw-Hill: New York. NY.1988. 6. Hawkins, J. M.: Meyer, A,:Lewis, T A,; Loren, S. D.; Hollander, F . J. Sciencr 1991. 252.312. 7. Ball, D.W J. Cham Edm. Submitted for publieation.
Volume 71 Number 6 June 1994
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